Poorly embeddable metric spaces and Group Theory Mikhail Ostrovskii - - PDF document

Poorly embeddable metric spaces and Group Theory

Mikhail Ostrovskii

• St. John’s University

Queens, New York City, NY e-mail: ostrovsm@stjohns.edu web page: http://facpub.stjohns.edu/ostrovsm March 2015, Geometric Group Theory on the Gulf Coast South Padre Island, Texas

• Introductory remarks:

Group Theory is an important tool for constructing differ- ent exotic combinatorial and metric struc-

• tures. In my educational talk I plan to de-

scribe constructions of metric spaces hav- ing poor embeddability properties into Ba- nach spaces, including basic tools used to prove poor embeddability and relevance of Group Theory.

• Throughout the talk I shall provide refer-

ences for those relevant results for which I do not plan to provide any details. Many of these references will be to my book “Met- ric Embeddings” (2013), which I shall cite as [ME]. Since the price of [ME] is rather large (≈$100), I would like to mention that you can get a free preliminary version at Google Scholar. (There were some small changes and some theorem numbers etc can be slightly different).

• Motivation: Why do we want to embed

metric spaces into Banach spaces? There are at least three reasons for this:

• (1) In Combinatorial Optimization it was

discovered that some problems, which were known to be computationally hard (the worst- case running time on an input of size n is known or believed to grow exponentially in n) admit fast (in the sense of worst-case running time) approximation algorithms which are constructed using embeddings of cer- tain metric spaces into such Banach spaces as ℓ2 (separable Hilbert space) and ℓ1 (the space of absolutely summable sequences with the norm - sum of absolute values).

• What is an approximation algorithm? If we

are looking, for example, for a largest set

• f pairwise adjacent vertices in a graph, a

1 2-approximation algorithm is an algorithm

which produces a set of pairwise adjacent vertices of size at least 1

2 of the maximal

possible.

• See [ME, Example 1.14 and Section 1.4]

for more. See the books of Vazirani, Ap- proximation Algorithms and Williamson-Shmoys The Design of Approx- imation Algorithms for much more.

• (2) In Topology and K-theory it was pre-

dicted by Gromov that some special cases

• f the Novikov and the Baum-Connes con-

jectures can be proved using coarse em- beddings of the corresponding groups into sufficiently good Banach spaces. The first success on these lines is due to Guoliang Yu (2000), after that there were many other successes on these lines.

• Do not ask me for more details on the

Baum-Connes conjecture, since so far I have not found a source containing an under- standable for me reasonably detailed de- scription of it of a reasonable length.

• (3) In Group Theory, Guentner and Kaminker

(2004) discovered that exactness of a group (Yu’s Property A) follows from sufficiently good embeddability of the group into a Hilbert space. Later this line of research was continued, see Nowak-Yu Large Scale Geometry, 2012, Section 5.9 and the cor- responding ‘Notes and Remarks’.

• Plan of this lecture: Since we are inter-

ested in ‘good’ embeddings it would be in- teresting to know what are the obstruc- tions to such embeddings. One of the main ways to get an obstruction is to prove a suitable Poincar´ e inequality. My lecture is devoted to Poincar´ e inequalities, their usage for establishing poor embeddability, and some ways of getting Poincar´ e inequal- ities using Group Theory.

• We start by introducing those Banach and

metric spaces which are important for this talk.

• Banach spaces of sequences:

In all of the spaces below xi ∈ R, addition of se- quences and their multiplication by scalars are defined componentwise. ℓp =

    {xi}∞

i=1 : ||{xi}∞ i=1|| =

 

∞

• i=1

|xi|p

 

1/p

< ∞

    ,

where 1 ≤ p < ∞. ℓ∞ =

• {xi}∞

i=1 : ||{xi}∞ i=1|| = sup i∈N

|xi| < ∞

• .

Observe that for p = 1 the formula for the ℓp-norm becomes simpler: ∞

i=1 |xi|.

• Banach spaces of functions:

Lp(0, 1), 1 ≤ p < ∞ is the space of those measurable functions on [0, 1] for which the Lebesgue integral

1

0 |f(x)|pdx is finite. Addition and

scalar multiplication are defined pointwise, the norm is given by ||f|| =

1

0 |f(x)|pdx

1/p.

Usually we omit (0, 1) from the notation of this space and denote it just Lp.

• Reminder: Norm on a linear space X is a

function ||·|| : X → R+ (where R+ is the set

• f all nonnegative real numbers) satisfying

the conditions: – Triangle inequality: ∀x, y ∈ X ||x+y|| ≤ ||x|| + ||y||. – Symmetry: ∀x ∈ X ∀α ∈ R ||αx|| = |α| · ||x|| – If x = 0, then ||x|| > 0.

• The fact that the norms introduced above

satisfy the triangle inequality requires non- trivial argument for 1 < p < ∞.

• Metric spaces: A metric space is a set X

endowed with a function d : X × X → R+ (where R+ is the set of all nonnegative real numbers) satisfying the conditions: – Triangle inequality: ∀x, y, z ∈ X d(x, z) ≤ d(x, y) + d(y, z). – Symmetry: ∀x, y ∈ X d(x, y) = d(y, x). – Separation axiom: ∀x, y ∈ X x = y ⇒ d(x, y) = 0. – ∀x ∈ X d(x, x) = 0.

• The function d is called a metric on X.

• Graphs with graph distances: Let G =

(V (G), E(G)) be a graph, so V is a set of

• bjects called vertices and E is some set
• f unordered pairs of vertices called edges.

We denote an unordered pair consisting of vertices u and v by uv and say that u and v are ends of uv.

• A walk in G is a finite sequence of the form

W = v0, e1, v1, e2, . . . , ek, vk whose terms are alternately vertices and edges such that, for 1 ≤ i ≤ k, the edge ei has ends vi−1 and

• vi. We say that W starts at v0 and ends at

vk, and that W is a v0vk-walk. The number k is called the lengths of the walk. A graph G is called connected if for each u, v ∈ V (G) there is a uv-walk in G.

• If G is connected, we endow V (G) with the

metric dG(u, v) = the length of the short- est uv-walk in G. The metric dG is called the graph distance. When we say “graph G with its graph distance” we mean the metric space (V (G), dG).

– Groups with word metrics: Let G be a finitely generated group and S be a finite generating set of G. We assume that S does not contain the identity and is symmetric, that is, contains g if and

• nly if it contains g−1.

– The Cayley graph of G corresponding to the generating set S is the graph whose vertices are elements of G, ele- ments g1 ∈ G and g2 ∈ G are connected by an edge if and only if g−1

1 g2 ∈ S.

– The graph distance of this graph is called the word metric because the distance between group elements g and h is the shortest representation of g−1h in terms

• f elements of S, such representations

are called words in the alphabet S.

• Each Banach space (and in particular all of

the spaces introduced above) is considered as a metric space with the metric d(x, y) = ||x − y||. The fact that the conditions for a metric space are satisfied by normed spaces follows from the definitions.

• Embeddings: By an embedding of a set

X into Y we mean any (not necessarily in- jective or surjective) map of X into Y .

• A map f : X → Y

between two metric spaces is called an isometric embedding if it preserves distances, that is dY (f(u), f(v)) = dX(u, v) for all u, v ∈ X.

• If there exists an isometric embedding of

X into Y we say that X is isometric to a subset (subspace) of Y .

• If an isometric embedding of X into Y is a

bijection of X and Y , we say that X and Y are isometric.

• The first observation which is worth men-

tioning right away is that for some Banach spaces there are no geometric obstructions for embeddability (of course there are al- ways some trivial obstructions of the type: we cannot embed a metric space having large cardinality into a Banach space hav- ing small cardinality).

• Proposition [Fr´

echet (1910)]. Each count- able metric space admits an isometric em- bedding into ℓ∞ (actually Fr´ echet proved this for a more general case of spaces ad- mitting countable dense set). – Proof: Let X = {ui}∞

i=0 be a countable

metric space. We introduce a map f : X → ℓ∞ by f(v) = {d(v, ui) − d(ui, u0)}∞

i=1.

Observe that ||f(v) − f(w)|| = sup

i∈N

|d(v, ui) − d(w, ui)|. – The triangle inequality implies sup

i∈N

|d(v, ui) − d(w, ui)| ≤ d(v, w). – On the other hand, if v = w, then at least one of v, w is among {ui}∞

i=1. Sup-

pose that v ∈ {ui}∞

i=1. We get

supi∈N |d(v, ui) − d(w, ui)| ≥ |d(v, v) − d(w, v)| = d(v, w). . If the space is bounded, the subtracted term is not needed.

• In many cases, in particular, in many im-

portant applications of embeddings there is no hope for isometric embeddings.

• For example, only very few graphs admit

isometric embeddings into ℓ2.

• Definition: A complete graph with n ver-

tices in which any two distinct vertices are joined by exactly one edge is denoted Kn. A path with n vertices is a graph whose vertices form a sequence {vi}n

i=1 and edges

are determined by the following: vk, k = 2, . . . , n − 1 is joined by exactly one edge with vk−1 and vk+1. The vertex v1 is joined with v2 only and the vertex vn is joined with vn−1 only. The path with n vertices is de- noted Pn. A graph is called simple if any two vertices in it are joined by at most one edge and there are no edges joining a ver- tex to itself. The degree of a vertex is the number of edges incident to it.

• Exercise [ME, Example 1.32]. A finite

simple connected graph G admits an iso- metric embedding into ℓ2 if and only if it is either Kn or Pn for some n.

• Hint. Prove and use the following fact: In

ℓ2 any three vectors satisfying ||x − z|| = ||x − y|| + ||y − z|| lie on the same line. (Ac- tually the same fact and the statement of the exercise is true for any ℓp and Lp with 1 < p < ∞, but there it takes more work to prove the result stated in the Hint.)

• Definition: Let C < ∞. A map f : (X, dX) →

(Y, dY ) between two metric spaces is called C-Lipschitz if ∀u, v ∈ X dY (f(u), f(v)) ≤ CdX(u, v). A map f is called Lipschitz if it is C-Lipschitz for some C < ∞. For a Lipschitz map f we define its Lipschitz constant by Lipf := sup

dX(u,v)=0

dY (f(u), f(v)) dX(u, v) .

• A wider class of embeddings.

Defini- tion: A map f : X → Y is called a C- bilipschitz embedding (or a C-bi-Lipschitz embedding) if there exists r > 0 such that rdX(u, v) ≤ dY (f(u), f(v)) ≤ rCdX(u, v) (1) for all u, v ∈ X. A bilipschitz embedding is an embedding which is C-bilipschitz for some C < ∞. The smallest constant C for which there exist r > 0 such that (1) is satisfied is called the distortion of f. (It is easy to see that such smallest constant exists.)

• It is easy to see that each bijective embed-

ding of a finite metric space is bilipschitz (possibly with very large distortion). So for bilipschitz embeddings of finite spaces the main focus is shifted to either finding low- distortion embeddings or finding bilipschitz embeddings of families of spaces with uni- formly bounded distortions.

• Let (X, dX) and (Y, dY ) be metric spaces.

The infimum of distortions of bilipschitz embeddings of X into Y is denoted cY (X). We let cY (X) = ∞ if there are no bilip- schitz embeddings of X into Y . When Y = Lp we use the notation cY (·) = cp(·) and call this number the Lp-distortion of

• X. The parameter c2(X) is called the Eu-

clidean distortion of X. Our next purpose is to develop some techniques for estimates

• f distortion cY (X) from below.
• We start with a simple example: consider a

4-cycle C4 and label its vertices in the cyclic

• rder: v1, v2, v3, v4. We are going to show

that the Euclidean distortion of C4 can be estimated using the following inequality ||f(v1) − f(v3)||2 + ||f(v2) − f(v4)||2 ≤ ||f(v1) − f(v2)||2 + ||f(v2) − f(v3)||2 +||f(v3) − f(v4)||2 + ||f(v4) − f(v1)||2, (2) which holds for an arbitrary collection f(v1), f(v2), f(v3), f(v4) of elements of a Hilbert space.

• To prove (2) we use the identity ||a−b||2 =

||a||2−2a, b+||b||2 for each of the terms in (2). Then we move everything to the right- hand side and observe that the obtained inequality can be written in the form 0 ≤ ||f(v1) − f(v2) + f(v3) − f(v4)||2.

• We postpone the computation of c2(C4)

slightly, introducing some terminology first. Inequality (2) can be considered as one of the simplest Poincar´ e inequalities for em- beddings of metric spaces.

• Definition: Let (X, dX) and (Y, dY ) be met-

ric spaces, Ψ : [0, ∞) → [0, ∞) be a non- decreasing function, au,v, bu,v, u, v ∈ X be arrays of nonnegative real numbers. If for an arbitrary function f : X → Y the in- equality

• u,v∈X

au,vΨ(dY (f(u), f(v))) ≥

• u,v∈X

bu,vΨ(dY (f(u), f(v))) (3) holds, we say that Y -valued functions on X satisfy the Poincar´ e inequality (3).

• Observe that in this inequality the struc-

ture of X plays no role, we use X just as a set of labels for elements f(u) ∈ Y .

• The inequality (3) is useful for the theory
• f embeddings only if a similar inequality

does not hold for the identical map on X, that is, if

• u,v∈X

au,vΨ(dX(u, v)) <

• u,v∈X

bu,vΨ(dX(u, v)). (4)

• In such a case we get immediately that X

is not isometric to a subset of Y .

• Definition: We call the quotient
• u,v∈X bu,vΨ(dX(u, v))
• u,v∈X au,vΨ(dX(u, v))

the Poincar´ e ratio of the metric space X corresponding to the Poincar´ e inequality (3) and denote it Pa,b,Ψ(t)(X).

• Having more information on the values of

sides of (4) and on the function Ψ, we can get an estimate for the distortion cY (X). The corresponding estimate of cY (X) is quite simple if Ψ(t) = tp for some p > 0.

• In fact, the following can be obtained by

simple manipulations with the definitions [ME, Proposition 4.3]: Proposition. If Y -valued functions on X satisfy the Poincar´ e inequality (3) with Ψ(t) = tp, then cY (X) ≥

• Pa,b,tp(X)

1/p .

(5)

• Now we are ready to estimate c2(C4).

It is clear that ||f(v1) − f(v3)||2 + ||f(v2) − f(v4)||2 ≤ ||f(v1)−f(v2)||2+||f(v2)−f(v3)||2+ ||f(v3) − f(v4)||2 + ||f(v4) − f(v1)||2 is a Poincar´ e inequality for ℓ2-valued func- tions on C4 (more precisely: for ℓ2-valued functions on V (C4)).

• The corresponding Poincar´

e ratio is: (dC4(v1, v3)2 + dC4(v2, v4)2)/ (dC4(v1, v2)2 + dC4(v2, v3)2 + dC4(v3, v4)2 + dC4(v4, v1)2) = 2.

• By the Proposition from the previous slide

we get c2(C4) ≥ √ 2.

• This estimate is sharp, this can be shown

by an embedding whose image is the set of all points in R2 with coordinates 0 and 1.

• It is known [ME, Theorem 4.14] that the

Lp-distortion of a finite metric space is equal to supremum of Poincar´ e ratios for all pos- sible choices of arrays (au,v) and (bu,v).

• Definition: For a graph G with vertex set

V and a subset F ⊂ V by ∂F we denote the set of edges connecting F and V \F. The expanding constant (a.k.a. Cheeger constant) of G is h(G) = inf{

|∂F| min{|F|,|V \F|} :

F ⊂ V, 0 < |F| < +∞} (where |A| denotes the cardi- nality of a set A.)

• Definition: A sequence {Gn} of graphs is

called a family of expanders if all of Gn are finite, connected, k-regular for some k ∈ N (this means that each vertex is inci- dent with exactly k edges), their expanding constants h(Gn) are bounded away from 0 (that is, there exists ε > 0 such that h(Gn) ≥ ε for all n), and their sizes (num- bers of vertices) tend to ∞ as n → ∞.

• The easiest and historically the first con-

structions of expanders were random graphs (Kolmogorov–Bardzin’ (1967), Pinsker (1973)).

• The first deterministic (not random) con-

structions of expanders were obtained by Margulis (1973) using Property (T) of groups introduced by Kazhdan (1967).

• I do not plan to introduce Property (T) in

this lecture, but I can describe expanders which we can obtain on this lines. Short introduction to Property (T) can be found in [ME, Sections 5.3 and 5.4]. If you want to learn more, I can recommen the book Bekka-de la Harpe-Valette Kazhdan’s prop- erty (T), 2008.

• Let S be a finite generating set in SL(3, Z)

and Sk be the corresponding generating sets in (finite) groups SL(3, Zk). Then the Cayley graphs of SL(3, Zk) with respect to Sk form a family of expanders.

• The following is a Poincar´

e inequality for L1-valued functions on a vertex set of a graph (it is useful mostly for expanders).

• We denote the adjacency matrix of a graph

G = (V, E) by {au,v}u,v∈V , that is au,v =

  

1 if u and v are adjacent

• therwise.

Let h be the expanding constant of G.

• Theorem (ME, Theorem 4.7): The fol-

lowing Poincar´ e inequality holds for L1-valued functions on V :

• u,v∈V

au,v||f(u)−f(v)|| ≥

• u,v∈V

h |V | ||f(u)−f(v)||. (6)

• This Poincar´

e inequality is very powerful. It and its versions for Lp-spaces (1 ≤ p < ∞) imply that the distortion of Lp-embeddings

• f k-regular expanders with n vertices and

Cheeger constant h is at least C h

p logk n,

where C is an absolute constant.

• It is known that the logarithmic distortion

is the largest possible. Bourgain (1985) proved that there exists an absolute con- stant C such c1(X) ≤ c2(X) ≤ C ln n for each n-element set X.

• It is important for us that the Poincar´

e in- equality for expanders implies that they re- sist even weaker, so-called coarse embed- dings into Lp or ℓp (1 ≤ p < ∞).

• Definition (Gromov): A map f : (X, dX) →

(Y, dY ) between two metric spaces is called a coarse embedding if there exist non-decrea- sing functions ρ1, ρ2 : [0, ∞) → [0, ∞) (ob- serve that this condition implies that ρ2 has finite values) such that limt→∞ ρ1(t) = ∞ and ∀u, v ∈ X ρ1(dX(u, v)) ≤ dY (f(u), f(v)) ≤ ρ2(dX(u, v)).

• The first problem about coarse embeddings

which is of importance for Topology was: find an infinite family of finite graphs with uniformly bounded degrees which are not uniformly coarsely embeddable into a Hilbert space (“uniformly” means that we cannot use the same ρ1 and ρ2 for all of them).

• It is well-known (in Banach space theory)

that a coarse embeddability into a Hilbert space is equivalent to the coarse embed- dability into L1 [ME, Observation 4.12]. So our next goal is to show that the Poincar´ e inequality (6) implies that expanders are not uniformly coarsely embeddable into L1.

• Assume the contrary, that is, there is an

embedding f : V → L1 satisfying ∀u, v ∈ V ρ1(dG(u, v)) ≤ ||f(u) − f(v)|| ≤ ρ2(dG(u, v))

• Combining this inequality with the Poincar´

e inequality for expanders (6) we get

• u,v∈V

h |V |ρ1(dG(u, v)) ≤

• u,v∈V

au,v||f(u) − f(v)|| ≤ k|V |ρ2(1).

• To estimate from below

u,v∈V h |V | ρ1(dG(u, v))

we observe that the number of vertices at distance D to a given vertex in a k-regular graph is at most 1+k+k(k−1)+· · ·+k(k−1)D−1 ≤ kD +1.

• Let D = logk
• |V |

2 − 1

• . Then there are at

most |V |

2

vertices with distance ≤ D to a given vertex. Therefore

• u,v∈V

h |V | ρ1(dG(u, v)) ≥ h |V | · |V |2 2 · ρ1

• logk
• |V |

2 − 1

• .

• Useful Poincar´

e inequalities arise from fixed point properties of isometric actions of groups

• n Banach spaces.
• We say that a group G acts (on the left)
• n a set X if for each g ∈ G and x ∈ X an

element gx ∈ X is defined (more formally, there is a map of G × X to X) such that g2(g1x) = (g2g1)x and 1x = x, where 1 is the identity element of G.

• Let X be a Banach space, denote by O(X)

the group of all linear isometries on X. A homomorphism π : G → O(X) is called a linear isometric representation.

• Group action can be described in a similar

fashion as a homomorphism of the group into the group of bijective maps of X. If G acts on X and x ∈ X, the set {gx : g ∈ G} is called the orbit of x.

• We consider actions of a group G by isome-

tries (=distance preserving bijective maps, that is (always) ||gx1−gx2|| = ||x1−x2||) on a Banach space X over R. By the Mazur- Ulam theorem, each such isometry can be (uniquely) represented as gx = π(g)x + b(g), where π is a linear isometry.

• Recall that the Mazur-Ulam Theorem (1932)

states that any distance preserving bijec- tion between two Banach spaces over R, which maps 0 to 0 is linear. See Nica (2012) for a very short proof.

• So picking b(g) in such a way that g0 −

b(g) = 0, we get that x → gx − b(g) is an isometry of X which maps 0 to 0. The Mazur-Ulam theorem implies that it is of the form π(g)x.

• Using the definition of the group action we

can get from here: – π is a linear isometric representation of the group G. – b(gh) = π(g)b(h) + b(g). The function b(g) is called a 1-cocycle (or just a co- cycle) for π (or with respect to π). The equation b(gh) = π(g)b(h)+b(g) is called a cocycle condition. – The converse is also true: if the func- tion b(g) satisfies the cocycle condition for the representation π, then gx = π(g)x+ b(g) defines a group action (by isome- tries).

• Pick any vector v0 ∈ X.

Then b(g) := π(g)v0−v0 defines a cocycle. Such cocycle is called a coboundary (or 1-coboundary). This cocycle corresponds to the action which can be described as “rotation about −v0”. In particular −v0 is a fixed point of the cor- responding group action.

• We use the following terminology: a fixed

point for a group action is a vector in X satisfying gx = x for all g ∈ G.

• It is easy to check that if an action has a

fixed point, then the corresponding cocycle is a coboundary.

• An example of an action without fixed points

is: Z acts on R by n(x) = x+n. This simple example can be generalized: If the group admits a surjective homomorphism τ onto Z and X is any Banach space, pick a vec- tor x0 ∈ X, x0 = 0, and define the action gx = x + τ(g)x0.

• It is interesting that there exist groups for

which each isometric action on a “suffi- ciently good” Banach space (say, on a Hilbert space) has a fixed point. Note: Each finitely generated infinite group admits an isometric action on ℓ∞ without fixed points.

• The study of such fixed point property for

Hilbert space was initiated (as far as I know) by Serre. If each isometric action of a group G on a Hilbert space has a fixed point, we say that G has property (FH).

• It turns out that (at least for finitely gen-

erated discrete groups, more generally, for σ-compact groups) this property coincides with the property (T) (result of Delorme (1977) and Guichardet (1972)) and so there are many groups having it, say, SL(n, Z) for n ≥ 3 (see the mentioned book Bekka- de la Harpe-Valette for more examples and for a proof of the equivalence of (T) and (FH)).

• The Poincar´

e inequality which I am going to present is more useful if stated in a more general context of relative property (FH).

• Definition.

A pair (G, H) (where H is a subgroup of G) has relative property (FH) if every isometric action of G on a Hilbert space is such that {g}g∈H have a common fixed point.

• Lemma.

If the pair (G, H) has relative Property (FH), then for every finite gener- ating subset S of G, there exists a constant C > 0 such that for every affine isometric action of G on ℓ2 the inequality ||b(g)||2 ≤ C

s∈S ||b(s)||2 holds for all g ∈ H.

– Proof. Assume on the contrary that there exists a sequence of isometric ac- tions of G on real Hilbert spaces {Hn}n with the corresponding cocycles {bn}, such that

s∈S ||bn(s)||2 ≤ 1 for all n ∈

N, and a sequence gn ∈ H such that ||bn(gn)|| → ∞ as n → ∞. Up to passing to a subsequence, we can assume that ||bn(gn)||2 > 2n.

– Consider the isometric action of G on the direct sum ℓ2 = (⊕nHn)2, corre- sponding to the direct sum of πn, and the 1-cocycle defined by b :=

n bn/n

(the cocycle condition is satisfied). As

• s∈S ||b(s)||2 ≤

n 1 n2 < ∞ and each g ∈

G is a finite product of elements of S, b is well defined. On the other hand, ||b(gn)||2 ≥ 2n/n2, hence b is not bounded

• n H, a contradiction because (as it is

easy to see) an action having a fixed point has a bounded cocycle.

• Proposition (Arzhantseva-Tessera (2014))

Let G be a finitely generated group, S be a finite generating set in G, and (G, H) has relative property (FH). Then there ex- ists C > 0 such that for all finite quo- tients Q of G, every function f from Q to a Hilbert space satisfies the following “rela- tive Poincar´ e” inequality: for every y ∈ H,

• g∈Q

||f(g¯ y)−f(g)||2 ≤ C

• g∈Q, s∈S

||f(g¯ s)−f(g)||2, where ¯ y and ¯ s denote the projection of y and s in Q.

• Proof: We apply the result of the Lemma

from the previous slide to the Hilbert space H = ℓ2(Q, ℓ2) of ℓ2-valued functions on Q, the representation π being right translation and the cocycle b(g) = f − π(g)f (the co- cycle of rotation about f), where f is an arbitrary function f : Q → ℓ2 (this function is in H since Q is finite).

• The Proposition from the previous slide im-

plies that finite quotients {Qn} of G, en- dowed with the metrics corresponding to the projections of S for which images of H do not have uniformly bounded diameters, are not uniformly coarsely embeddable into a Hilbert space.

• Proof:

Assume the contrary. Let fn : Qn → ℓ2 be uniformly coarse embeddings. By the Proposition we get

• g∈Qn

||fn(g¯ y) − fn(g)||2 ≤ C

• g∈Qn, s∈S

||fn(g¯ s) − fn(g)||2, The definition of uniformly coarse embed- dings gives us ρ1(dCay(Qn,¯

S)(¯

z, 1)) ≤ ||fn(g¯ z) − fn(g)|| ≤ ρ2(dCay(Qn,¯

S)(¯

z, 1)), where z is any element in G and ¯ z is its image in Qn.

• Combining these inequalities we get that

for every y ∈ H the following inequality holds: |Qn|(ρ1(dCay(Qn,¯

S)(¯

y, 1)))2 ≤ C|Qn||S|(ρ2(1))2

• If the distances dCay(Qn,¯

S)(¯

y, 1) are not uni- formly bounded over y ∈ H, we get a con- tradiction.

• Arzhantseva and Tessera (2014) used this

inequality and a very nice group-theoretical construction to answer in the negative the following problem:

• Problem [ME Problem 11.9]: Suppose

that a metric space M with bounded ge-

• metry is not coarsely embeddable into ℓ2.

Does it follow that M weakly contains a family of expanders? (If you do not know what bounded geometry means, do not worry: it was enough to find a counterexample among infinite graphs with uniformly bounded degrees.)

• Open problem (Yu (2006)): Let p > q ≥
• 2. Construct a discrete metric space Γ with

bounded geometry which admits a coarse embedding into ℓp but not into ℓq.

• If you do not know what bounded geome-

try means, do not worry: restriction of this problem to infinite graphs with uniformly bounded degree is as interesting (and, most probably, is equivalent to the original prob- lem).

• The goal of this problem is to understand:

are there any cases for which the Kasparov- Yu (2006) results (based on coarse em- beddability into a uniformly convex space) are applicable, but the earlier results of Yu (2000) (based on a coarse embeddability into a Hilbert space) are not.

• For this problem analogues of the prop-

erty (FH) for other Banach spaces could be useful.

• The analogue of the definition of property

(FH) for other Banach spaces is obvious.

• A systematic study of this property started

in the paper of Bader, Furman, Gelander, and Monod (2007), where they proved that (FLp) for p ∈ (1, 2] is equivalent to (FH).

• They also proved that SL(n, Z) with n ≥ 3

has (FLp) for all p ∈ (1, ∞). However only very small amount of information is avail- able about the analogue of the Property (FH) even for Lp, p ∈ (2, ∞). In particu- lar, the analogues of most of the results used by Arzhantseva-Tessera in the men- tioned paper are either false or unknown for this case. One of the exceptions is the Proposition about the Poincar´ e inequality (proved above) which admits an immediate generalization to the Lp-case.